Isn't a High Speed Line from Edinburgh to Glasgow just a side effect of the need for the lines from each to Carlisle to meet somewhere, much like the only electrified route used to be.
But no one knows where this junction is going to be. It could well be much further South of Carstairs, which makes it a longer route. And remember that Carstairs Junction is painfully slow going between Glasgow and Edinburgh. I suspect that any such junction would be designed with Anglo-Scottish journeys in mind, and any intra-Scottish routes would be a secondary concern.
So, we could potentially have a junction that allows a useful high speed rail link. We could also have a junction that provides a secondary line which, whilst travelling at High Speed, will take longer than conventional rail links for various reasons.
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I think the junction you are referring to is Greenhill Junction, Bonnybridge, where the connecting chord diverges on to the Caledonian Railway main line northbound towards Stirling.
That's the name I couldn't remember! Thanks!
So you could increase capacity by having 4-track between Glasgow and Greenhill Jn with the slow tracks going elsewhere (possibly onto our fictional Crossrail and through to Ayrshire?). I think 4-track could be justified on this stretch, although I'm not sure if it's possible. (A quick look at google earth makes Bishopbriggs look tight; Lenzie is possible but you'd need to rebuild the local road, Croy looks easy enough. Major problems getting around the Castlecary viaduct, mind...)
I suspect that, given the scope of the current work, this is a long way off if it ever happens. Castlecary alone is a barrier, and given that Welwyn viaduct proves a similar barrier on the ECML, I don't think we'll see this. But perhaps Lenzie-Croy 4-tracking could provide some additional capacity?